Can I use neural networks for a problem (in description)?
To answer the titular question first: Yes, of course you can. Whether a neural network can give you better results than a simpler model, however, depends on:


How complex/non-linear the relationship between your input and output variables is;
Whether the neural network you have specified is able to learn this relationship efficiently;
How much training data you have.


With the information you have provided in the question, there's really no telling how it will perform against a simpler model, so I would advice trying both. 

As for how your output varies over time and location, just include both in your model. In case of a linear regression model, accounting for spatiotemporal autocorrelation could be done with a mixed model using an appropriate covariance structure. The challenge for a neural network would be how to specify one that can learn this type of relationship (for starters, read up on RNNs for longitudinal data).